Simplicial faces of the set of correlation matrices

Abstract

This paper concerns the facial geometry of the set of $n\times n$ correlation matrices. The main result states that almost every set of $r$ vertices generates a simplicial face, provided that $r\le \sqrt{\mathrm{c}n}$, where $\mathrm{c}$ is an absolute constant. This bound is qualitatively sharp because the set of correlation matrices has no simplicial face generated by more than $\sqrt{2n}$ vertices.

12 pages, 2 figures

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